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. 1999 Feb;76(2):768–781. doi: 10.1016/S0006-3495(99)77242-5

(In)validity of the constant field and constant currents assumptions in theories of ion transport.

A Syganow 1, E von Kitzing 1
PMCID: PMC1300080  PMID: 9929480

Abstract

Constant electric fields and constant ion currents are often considered in theories of ion transport. Therefore, it is important to understand the validity of these helpful concepts. The constant field assumption requires that the charge density of permeant ions and flexible polar groups is virtually voltage independent. We present analytic relations that indicate the conditions under which the constant field approximation applies. Barrier models are frequently fitted to experimental current-voltage curves to describe ion transport. These models are based on three fundamental characteristics: a constant electric field, negligible concerted motions of ions inside the channel (an ion can enter only an empty site), and concentration-independent energy profiles. An analysis of those fundamental assumptions of barrier models shows that those approximations require large barriers because the electrostatic interaction is strong and has a long range. In the constant currents assumption, the current of each permeating ion species is considered to be constant throughout the channel; thus ion pairing is explicitly ignored. In inhomogeneous steady-state systems, the association rate constant determines the strength of ion pairing. Among permeable ions, however, the ion association rate constants are not small, according to modern diffusion-limited reaction rate theories. A mathematical formulation of a constant currents condition indicates that ion pairing very likely has an effect but does not dominate ion transport.

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Selected References

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  1. Adams D. J., Dwyer T. M., Hille B. The permeability of endplate channels to monovalent and divalent metal cations. J Gen Physiol. 1980 May;75(5):493–510. doi: 10.1085/jgp.75.5.493. [DOI] [PMC free article] [PubMed] [Google Scholar]
  2. Chen D. P., Eisenberg R. S. Flux, coupling, and selectivity in ionic channels of one conformation. Biophys J. 1993 Aug;65(2):727–746. doi: 10.1016/S0006-3495(93)81099-3. [DOI] [PMC free article] [PubMed] [Google Scholar]
  3. Chen D., Lear J., Eisenberg B. Permeation through an open channel: Poisson-Nernst-Planck theory of a synthetic ionic channel. Biophys J. 1997 Jan;72(1):97–116. doi: 10.1016/S0006-3495(97)78650-8. [DOI] [PMC free article] [PubMed] [Google Scholar]
  4. Cheng W., Wang C. X., Chen W. Z., Xu Y. W., Shi Y. Y. Investigating the dielectric effects of channel pore water on the electrostatic barriers of the permeation ion by the finite difference Poisson-Boltzmann method. Eur Biophys J. 1998;27(2):105–112. doi: 10.1007/s002490050116. [DOI] [PubMed] [Google Scholar]
  5. Conti F., Eisenman G. The steady-state properties of an ion exchange membrane with mobile sites. Biophys J. 1966 May;6(3):227–246. doi: 10.1016/S0006-3495(66)86653-5. [DOI] [PMC free article] [PubMed] [Google Scholar]
  6. Cooper K., Jakobsson E., Wolynes P. The theory of ion transport through membrane channels. Prog Biophys Mol Biol. 1985;46(1):51–96. doi: 10.1016/0079-6107(85)90012-4. [DOI] [PubMed] [Google Scholar]
  7. Dorman V., Partenskii M. B., Jordan P. C. A semi-microscopic Monte Carlo study of permeation energetics in a gramicidin-like channel: the origin of cation selectivity. Biophys J. 1996 Jan;70(1):121–134. doi: 10.1016/S0006-3495(96)79554-1. [DOI] [PMC free article] [PubMed] [Google Scholar]
  8. Doyle D. A., Morais Cabral J., Pfuetzner R. A., Kuo A., Gulbis J. M., Cohen S. L., Chait B. T., MacKinnon R. The structure of the potassium channel: molecular basis of K+ conduction and selectivity. Science. 1998 Apr 3;280(5360):69–77. doi: 10.1126/science.280.5360.69. [DOI] [PubMed] [Google Scholar]
  9. Eisenberg R. S. Computing the field in proteins and channels. J Membr Biol. 1996 Mar;150(1):1–25. doi: 10.1007/s002329900026. [DOI] [PubMed] [Google Scholar]
  10. Furois-Corbin S., Pullman A. The effect of point mutations on energy profiles in a model of the nicotinic acetylcholine receptor (AChR) channel. Biophys Chem. 1991 Feb;39(2):153–159. doi: 10.1016/0301-4622(91)85017-k. [DOI] [PubMed] [Google Scholar]
  11. Goldman D. E. POTENTIAL, IMPEDANCE, AND RECTIFICATION IN MEMBRANES. J Gen Physiol. 1943 Sep 20;27(1):37–60. doi: 10.1085/jgp.27.1.37. [DOI] [PMC free article] [PubMed] [Google Scholar]
  12. HODGKIN A. L., KATZ B. The effect of sodium ions on the electrical activity of giant axon of the squid. J Physiol. 1949 Mar 1;108(1):37–77. doi: 10.1113/jphysiol.1949.sp004310. [DOI] [PMC free article] [PubMed] [Google Scholar]
  13. Horn R. Run, don't hop, through the nearest calcium channel. Biophys J. 1998 Sep;75(3):1142–1143. doi: 10.1016/S0006-3495(98)74033-0. [DOI] [PMC free article] [PubMed] [Google Scholar]
  14. Imoto K., Busch C., Sakmann B., Mishina M., Konno T., Nakai J., Bujo H., Mori Y., Fukuda K., Numa S. Rings of negatively charged amino acids determine the acetylcholine receptor channel conductance. Nature. 1988 Oct 13;335(6191):645–648. doi: 10.1038/335645a0. [DOI] [PubMed] [Google Scholar]
  15. Klement R., Soumpasis D. M., Kitzing E. V., Jovin T. M. Inclusion of ionic interactions in force field calculations of charged biomolecules--DNA structural transitions. Biopolymers. 1990 May-Jun;29(6-7):1089–1103. doi: 10.1002/bip.360290620. [DOI] [PubMed] [Google Scholar]
  16. Konno T., Busch C., Von Kitzing E., Imoto K., Wang F., Nakai J., Mishina M., Numa S., Sakmann B. Rings of anionic amino acids as structural determinants of ion selectivity in the acetylcholine receptor channel. Proc Biol Sci. 1991 May 22;244(1310):69–79. doi: 10.1098/rspb.1991.0053. [DOI] [PubMed] [Google Scholar]
  17. Levitt D. G. Exact continuum solution for a channel that can be occupied by two ions. Biophys J. 1987 Sep;52(3):455–466. doi: 10.1016/S0006-3495(87)83234-4. [DOI] [PMC free article] [PubMed] [Google Scholar]
  18. Levitt D. G. Interpretation of biological ion channel flux data--reaction-rate versus continuum theory. Annu Rev Biophys Biophys Chem. 1986;15:29–57. doi: 10.1146/annurev.bb.15.060186.000333. [DOI] [PubMed] [Google Scholar]
  19. MacGillivray A. D., Hare D. Applicability of Goldman's constant field assumption to biological systems. J Theor Biol. 1969 Oct;25(1):113–126. doi: 10.1016/s0022-5193(69)80019-6. [DOI] [PubMed] [Google Scholar]
  20. Macias F., Starzak M. E. Ion velocity distributions in gramicidin channels determined with laser Doppler velocimetry. Biochim Biophys Acta. 1993 Dec 12;1153(2):331–334. doi: 10.1016/0005-2736(93)90423-w. [DOI] [PubMed] [Google Scholar]
  21. McGill P., Schumaker M. F. Boundary conditions for- single-ion diffusion. Biophys J. 1996 Oct;71(4):1723–1742. doi: 10.1016/S0006-3495(96)79374-8. [DOI] [PMC free article] [PubMed] [Google Scholar]
  22. Muller R. U., Finkelstein A. The effect of surface charge on the voltage-dependent conductance induced in thin lipid membranes by monazomycin. J Gen Physiol. 1972 Sep;60(3):285–306. doi: 10.1085/jgp.60.3.285. [DOI] [PMC free article] [PubMed] [Google Scholar]
  23. Nonner W., Chen D. P., Eisenberg B. Anomalous mole fraction effect, electrostatics, and binding in ionic channels. Biophys J. 1998 May;74(5):2327–2334. doi: 10.1016/S0006-3495(98)77942-1. [DOI] [PMC free article] [PubMed] [Google Scholar]
  24. Nonner W., Eisenberg B. Ion permeation and glutamate residues linked by Poisson-Nernst-Planck theory in L-type calcium channels. Biophys J. 1998 Sep;75(3):1287–1305. doi: 10.1016/S0006-3495(98)74048-2. [DOI] [PMC free article] [PubMed] [Google Scholar]
  25. Parsegian A. Energy of an ion crossing a low dielectric membrane: solutions to four relevant electrostatic problems. Nature. 1969 Mar 1;221(5183):844–846. doi: 10.1038/221844a0. [DOI] [PubMed] [Google Scholar]
  26. Parsegian V. A. Ion-membrane interactions as structural forces. Ann N Y Acad Sci. 1975 Dec 30;264:161–171. doi: 10.1111/j.1749-6632.1975.tb31481.x. [DOI] [PubMed] [Google Scholar]
  27. Partenskii M. B., Dorman V., Jordan P. C. Influence of a channel-forming peptide on energy barriers to ion permeation, viewed from a continuum dielectric perspective. Biophys J. 1994 Oct;67(4):1429–1438. doi: 10.1016/S0006-3495(94)80616-2. [DOI] [PMC free article] [PubMed] [Google Scholar]
  28. Pullman A. Energy profiles in the gramicidin A channel. Q Rev Biophys. 1987 Nov;20(3-4):173–200. doi: 10.1017/s0033583500004170. [DOI] [PubMed] [Google Scholar]
  29. Ranatunga K. M., Kerr I. D., Adcock C., Smith G. R., Sansom M. S. Protein-water-ion interactions in a model of the pore domain of a potassium channel: a simulation study. Biochim Biophys Acta. 1998 Mar 6;1370(1):1–7. doi: 10.1016/s0005-2736(97)00271-x. [DOI] [PubMed] [Google Scholar]
  30. Roux B., Karplus M. Molecular dynamics simulations of the gramicidin channel. Annu Rev Biophys Biomol Struct. 1994;23:731–761. doi: 10.1146/annurev.bb.23.060194.003503. [DOI] [PubMed] [Google Scholar]
  31. Roux B., Prod'hom B., Karplus M. Ion transport in the gramicidin channel: molecular dynamics study of single and double occupancy. Biophys J. 1995 Mar;68(3):876–892. doi: 10.1016/S0006-3495(95)80264-X. [DOI] [PMC free article] [PubMed] [Google Scholar]
  32. Roux B. Valence selectivity of the gramicidin channel: a molecular dynamics free energy perturbation study. Biophys J. 1996 Dec;71(6):3177–3185. doi: 10.1016/S0006-3495(96)79511-5. [DOI] [PMC free article] [PubMed] [Google Scholar]
  33. Sansom M. S., Smith G. R., Adcock C., Biggin P. C. The dielectric properties of water within model transbilayer pores. Biophys J. 1997 Nov;73(5):2404–2415. doi: 10.1016/S0006-3495(97)78269-9. [DOI] [PMC free article] [PubMed] [Google Scholar]
  34. Smith G. R., Sansom M. S. Molecular dynamics study of water and Na+ ions in models of the pore region of the nicotinic acetylcholine receptor. Biophys J. 1997 Sep;73(3):1364–1381. doi: 10.1016/S0006-3495(97)78169-4. [DOI] [PMC free article] [PubMed] [Google Scholar]
  35. Soumpasis D. M. Statistical mechanics of the B----Z transition of DNA: contribution of diffuse ionic interactions. Proc Natl Acad Sci U S A. 1984 Aug;81(16):5116–5120. doi: 10.1073/pnas.81.16.5116. [DOI] [PMC free article] [PubMed] [Google Scholar]
  36. Unwin N. Acetylcholine receptor channel imaged in the open state. Nature. 1995 Jan 5;373(6509):37–43. doi: 10.1038/373037a0. [DOI] [PubMed] [Google Scholar]
  37. Villarroel A., Sakmann B. Threonine in the selectivity filter of the acetylcholine receptor channel. Biophys J. 1992 Apr;62(1):196–208. doi: 10.1016/S0006-3495(92)81805-2. [DOI] [PMC free article] [PubMed] [Google Scholar]
  38. von Kitzing E., Soumpasis D. M. Electrostatics of a simple membrane model using Green's functions formalism. Biophys J. 1996 Aug;71(2):795–810. doi: 10.1016/S0006-3495(96)79281-0. [DOI] [PMC free article] [PubMed] [Google Scholar]

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